Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF¡Çs.
There is a slight difference between the Gel¡Çfand-Kapranov-Zelevinski
HGF¡Çs and the Horn HGF¡Çs. The latter may
contain so called ¡Èpersistent polynomial solutions¡É that cannot be mapped
to GKZ HGF¡Çs via a natural isomorphism
between two spaces of HGF¡Çs.
In this talk, I will review basic facts on the Horn HGF¡Çs. As a main tool
to study the topology of the discriminant loci together with the analytic
aspects of the story, amoebas ? image by the log map of the discriminant-
will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.
Seminar on Analysis at University of Tsukuba